Introduction History Topological Games Recent Results

Semitopological groups versus topological groups

Warren B. Moors

The University of Auckland

August 24, 2015

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Table of Contents

1 Introduction

2 History

3 Topological Games

4 Recent Results

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Semitopological groups

A triple (G, ·,τ) is called a semitopological group if: (i) (G, ·) is a group and (G,τ) is a topological space; (ii) multiplication, (x, y) 7→ x · y, from G × G into G is separately continuous.

A triple (G, ·,τ) is a topological group if: (i) (G, ·) is a group and (G,τ) is a topological space; (ii) multiplication, (x, y) 7→ x · y, from G × G into G is jointly continuous; (iii) inversion, x 7→ x −1, from G onto G is continuous.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Clearly, every topological group is a semitopological group. However, there are some other interesting/natural examples that are not topological groups.

Example 1. (R, +,τS), where τS (the Sorgenfrey topology) is the topology on R generated by the sets

{[a, b) : a, b ∈ R and a < b}. Note that in this example (x, y) 7→ x · y is continuous but x 7→ x −1 is not. Example 2. Let (X,τ) be a nonempty topological space and let G be a nonempty subset of X X . If (G, ◦) is a group (where “◦" denotes the binary relation of function composition) and τp denotes the topology on X X of pointwise convergence on X then (G, ◦,τp) is a semitopological group provided the members of G are continuous functions. Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Example 3. Let G denote the set of all homeomorphisms on (R,τS). From Example 2 we see that (G, ◦,τp) is a semi topological group. However, (G, ◦,τp) is not a topological group.

To see this, define gn : R → R by, gn(x) := [1 + 1/(n + 1)]x, an := 1 + 1/(2n) and

x if x 6∈ [an, an + 1/(2n)) ∪ [n, n + 1/(2n))  fn(x) :=  n + (x − an) if x ∈ [an, an + 1/(2n)) an + (x − n) if x ∈ [n, n + 1/(2n)).  Then both fn and gn converge pointwise to id - the identity map, however, lim (fn ◦ gn)(1)= ∞= 6 (id ◦ id)(1)= id(1)= 1. n→∞

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

On the other hand sometimes Example 2 does give rise to topological groups. Example 4. Let (M, d) be a metric space and let G be the set of all isometries on (M, d). Then (G, ◦,τp) is a topological group. To prove this, one just needs to apply the triangle inequalitya few times. Indeed, if fn, gn ∈ G, fn converges pointwise to f ∈ G and gn converges pointwise to g ∈ G then for each x ∈ M,

0 ≤ d(fn(gn(x)), f (g(x)))

≤ d(fn(gn(x)), fn(g(x)) + d(fn(g(x)), f (g(x)))

= d(gn(x), g(x)) + d(fn(g(x)), f (g(x)).

Semitopological groups also naturally arise in the study of group actions (topological dynamics).

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Example 5. Let (G, ·) be a group and let (X,τ) be a topological space. Further, let π : G × X → X be a mapping (group action) such that: (i) π(e, x)= x for all x ∈ X, where e denotes the identity element of G; (ii) π(g · h, x)= π(g,π(h, x)) for all g, h ∈ G and x ∈ X; (iii) for each g ∈ G, the mapping, x 7→ π(g, x), is a continuous function on X. Then (G, X) is called a flow on X. If we consider the mapping ρ : G → X X defined by, ρ(g)(x)= π(g, x) for all x ∈ X.

Then (ρ(G), ◦,τp) is a semitopological group.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Table of Contents

1 Introduction

2 History

3 Topological Games

4 Recent Results

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Research on the problem of which topological conditions on a semitopological group imply that it is a topological group possibly began in [D. Montgomery,“Continuity in topological groups" Bull. Amer. Math. Soc. 42 (1936)] when the author showed that each completely metrizable semitopological group has jointly continuous multiplication. Later, in [R. Ellis,“A note on the continuity of the inverse" Proc. Amer. Math. Soc. 8 (1957) and “Locally compact transformation groups" Duke Math. J. 24 (1957)] Ellis showed that each locally compact semitopological group is in fact a topological group. This answered a question raised by A. D. Wallace in

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results [A. D. Wallace,“The structure of topological semigroups" Bull. Amer. Math. 61 (1955)]. Next in

[W. Zelazko,“A theorem on B0 division algebras" Bull. Acad. Pol. Sci. 8 (1960)] Zelazko used Montgomery’s result from 1936 to show that each completely metrizable semitopological group is a topological group. Much later, in [A. Bouziad,“Every Cech-analyticˇ Baire semitopological group is a topological group" Proc. Amer. Math. Soc. 124 (1996)] Bouziad improved both of these results and answered a question raised by Pfister in [H. Pfister,“Continuity of the inverse" Proc. Amer. Math. Soc. 95 (1985)]

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results by showing that each Cech-completeˇ semitopological group is a topological group. (Recall that a topological space (X,τ) is called Cech-complete if it is a Gδ subset of a compact Hausdorff space.) It is well-known that both locally compact and completely metrizable topological spaces are Cech-complete).ˇ To do this, it was sufficient for Bouziad to show that every Cech-completeˇ semitopological group has jointly continuous multiplication since earlier, Brand [N. Brand,“Another note on the continuity of the inverse" Arch. Math. 39 (1982)] had proven that every Cech-completeˇ semitopological group with jointly continuous multiplication is a topological group. Brand’s proof of this was later improved and simplified in

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

[H. Pfister,“Continuity of the inverse" Proc. Amer. Math. Soc. 95 (1985)]. Apart from those named above there have been many other contributors to the question of when a semitopological group is in fact a topological group. For example, Arhangel’skii, Brown, Cao, Choban, Drozdowski, Guran, Hola, Kenderov, Korovin, Kortezov, Lawson, Moors, Namioka, Piotrowski, Ravsky, Reznichenko, Romaguera, Sanchis and Tkachenko.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Table of Contents

1 Introduction

2 History

3 Topological Games

4 Recent Results

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Topological Games

In [P. S. Kenderov, I. Kortezov and W.B. Moors,“Topological games and topological groups" Topology Appl. 109 (2001)] the authors used a two player topological game to determine some conditions on a semitopological group that imply it is a topological group. Using this game they were able to prove a theorem considerably more general than the following. Theorem 1. Let (G, ·,τ) be a semitopological group such that (G,τ) is a regular Baire space. If any of the following conditions hold, then (G, ·,τ) is a topological group. (i) (G,τ) is metrizable (or more generally, (G,τ) is a p-space); (ii) (G,τ) is Cech-analyticˇ (or more generally, has countable separation); (iii) (G,τ) is locally countably compact.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Recall that a topological space (X,τ) is called (i) regular if every closed subset of X and every point outside of this set, can be separated by disjoint open sets and (ii) Baire if the intersection of every countable family of dense open sets is dense in X. An advantage to the “game" approach used in [KKM] is that it covers many different situations at once. However, there is also a disadvantage to the game approach. Namely, people find the use of games unappealing, artificial and hard to understand. Hence some of the consequences of the paper [KKM] have gone unnoticed until quite recently. However, there is now a cottage industry showing that certain topological spaces satisfy the game hypotheses given in [KKM]. Unfortunately, most of these results were already known to the authors of [KKM] back in 2001. Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

The game that we shall consider involves two players which we will call player α and player β. The “field/court" that the game is played on is a fixed topological space (X,τ) with a fixed dense subset D. Thenameofthegameisthe“GS(D)-game". After naming the game we need to describe how to “play" the GS(D)-game. The player labeled β starts the game every time (life is not always fair). For their first move the player β must select a nonempty open subset B1 of X. Next, α gets a turn. For α’s first move he/she must select a nonempty open subset A1 of B1. This ends the first round of the game. In the second round, β goes first (again) and selects a nonempty open subset B2 of A1.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

α then gets to respond by choosing a nonempty open subset A2 of B2. This ends the second round of the game. At this stage we have A2 ⊆ B2 ⊆ A1 ⊆ B1. In general, after α and β have played the first n-rounds of the GS(D)-game, β will have selected nonempty open sets B1, B2,..., Bn and α will have selected nonempty open sets A1, A2,..., An such that:

An ⊆ Bn ⊆ An−1 ⊆ Bn−1 ⊆ · · · ⊆ A2 ⊆ B2 ⊆ A1 ⊆ B1.

At the start of the (n + 1)-round of the game, β goes first (again!) and selects a nonempty open subset Bn+1 of An. As with the previous n-rounds, player α gets to respond to this move by selecting a nonempty open subset An+1 of Bn+1.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Continuing this process indefinitely (i.e., continuing-on forever) the players produce an infinite sequence, (called a play of the GS(D)-game) {(An, Bn) : n ∈ N} of pairs of nonempty open subsets of X such that

An+1 ⊆ Bn+1 ⊆ An ⊆ Bn for all n ∈ N.

As with any game, we need a rule to determine who wins – otherwise it is a very boring game. ✂❦✁

We shall declare that α wins a play {(An, Bn) : n ∈ N} of the GS(D)-game if:

(i) ∈N An 6= ∅ and Tn (ii) each sequence (xn : n ∈ N) in D with xn ∈ An for all n ∈ N, has a cluster-point in X.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

If α does not win a play of the GS(D)-game then we declare that β wins that play of the GS(D)-game. So every play is won by either α or β and no play is won by both players. Continuing further into game theory we need to introduce the notion of a strategy. A strategy for the player β (player α) is a “rule" that specifies how the player β (player α) must respond/move in every possible situation that may occur during the course of the game. [A more precise mathematical description of a strategy is possible, but we shall not give it here.] We may now finally define a “strongly Baire" space.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

We shall say that a topological space (X,τ) is strongly Baire if it is regular and there exists a dense subset D of X such that the player β (i.e., the player with the privilege of going first) does NOT have a winning strategy in the GS(D)-game played on X (that is to say, that no matter what strategy player β adopts there is always at least one play of the GS(D)-game where α wins.). Clearly, if α actually possesses a winning strategy himself/ herself then β cannot possibly possess a winning strategy as well and so all spaces (X,τ) in which α has a winning strategy in the GS(D)-game are strongly Baire. Note: significantly there are some strongly Baire spaces in which the player α does not possess a winning strategy.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Structure of Game Proofs

Using the notion of a strongly Baire space the authors in [KKM] proved the following very general result. Theorem 2. [KKM, 2001] Let (G, ·,τ) be a semitopological space. If (G,τ) is a strongly Baire space then (G, ·,τ) is a topological group. The use of games can often simplify the presentation of certain inductive arguments. One can design a game that exactly suits/fits the particular inductive argument under consideration. That is, the game can be tailor made to fit the situation. The proof then divides into two parts. In one part we use the tailor made game to expedite the proof of the inductive argument. Strategies offering an effect way of recording the inductive hypotheses.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

The other part of the proof is then to determine those spaces/situations where the game conditions are satisfied. This dividing of the proof into two parts is an important feature of the game approach. Another feature of the game formalism is the possibility of considering spaces where neither player possesses a winning strategy. Initially, it is not at all clear how one can use the assumption:

“I do not possess a winning strategy.”

The way in which one usually exploits the hypothesis/condition that β does not possess a winning strategy is the following. One uses a proof by contradiction. That is, assume that the conclusion of the statement (that one wants to prove) is false.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Then use this additional information to construct a strategy t for the player β. The fact that t is not a winning strategy for the player β then yields the existence of a play {(An, Bn) : n ∈ N} where α wins.

This play {(An, Bn) : n ∈ N} is then used to obtain the required contradiction.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results Table of Contents

1 Introduction

2 History

3 Topological Games

4 Recent Results

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

As mentioned earlier, the difficulty with the game formulation is that many people that are not familiar with game theory are intimidated. The net result is that many of the consequences of the paper [KKM] have gone unnoticed - though this is starting to change now. As an example: one can easily show that every Baire metric space is a strongly Baire space. Thus, we have that every Baire metric semitopological group is a topological group. However, even in [M. Tkachenko, “Paratopological and semitopological groups versus topological groups" Recent Progress in General Topology. III, Atlantis Press, Paris, 2014] the author says:

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

“Further, Solecki and Srivastava establish in [91] a general fact implying that every separable metrizable Baire semitopological group is a topological group and mention there that, by an unpublished result of Reznichenko, one can drop the separability restriction in this corollary. Let us show, using an argument from [80] - (an unpublished manuscript), that this is certainly the case.” By modifying the game considered in [KKM] one can further generalise the class of strongly Baire spaces. This was done in both W. Moors, “Semitopological groups, Bouziad spaces and topological groups” Topology Appl. 160 (2013). and

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

W. Moors, “Any semitopological group that is homeomorphic to a product of Cech-complete spaces is a topological group” Set-Valued Var. Anal. 21 (2013). to solve some open questions. In particular, the above papers solve most of the open problems in [M. Tkachenko, “Paratopological and semitopological groups versus topological groups" Recent Progress in General Topology. III, Atlantis Press, Paris, 2014] (concerning when a semitopological group is a to topological group) as well as ALL the open problems in [A. Arhangel’skii, M. Choban, and P. Kenderov, “Topological games and topologies on groups” Math. Maced. 8 (2010)].

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Thank you for your attention and for the opportunity to present my work.

A PDF version of this talk is available at:

www.math.auckland.ac.nz/∼moors/

——————————– The End ——————————–

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results But wait, there is more...

Although the class of strongly Baire spaces provided a convenient framework for our theorems these spaces are, unfortunately, not readily identifiable. So in this part of the talk we will introduce a related class of spaces whose membership properties are more readily determined. [This is the second part of the game approach that was mentioned earlier]. A topological space (Y ,τ) is said to be cover semi-complete if there exists a pseudo-metric d on Y such that; (i) each d-convergent sequence in Y has a cluster-point in Y ; (ii) (ii) Y is fragmented by d, that is, for each ε> 0 and nonempty subset A of Y there exists a nonempty relatively open subset B of A such that d-diam B < ε.

Warren B. Moors Semitopological groups versus topological groups Introduction History Topological Games Recent Results

Clearly, all metric spaces are cover semi-complete, as are, all regular countably compact spaces. With a little more work one can show that all Cech-analytic spaces (which includes all Cech-complete spaces) are cover semi-complete. THEOREM 3. [KKM, 2001] Let (G, ·,τ) be a semitopological group. If (G,τ) contains, as a second category subset, a cover semi-complete space Y , then (G, ·,τ) is a topological group. In particular, if (G,τ) is a cover semi-complete Baire space then (G, ·,τ) is a topological group. ——————————– The End ——————————–

Warren B. Moors Semitopological groups versus topological groups